Schwarz (1975) theorem

theorem:

Let $\mathrm{\Gamma }$ be a compact Lie group acting on $V$. Let $u_1,\mathrm{\dots },u_s$ be a Hilbert basis for the $\mathrm{\Gamma }$-invariant polynomials $𝒫(\mathrm{\Gamma })$ (see Hilbert-Weyl theorem). Let $f\in ℰ(\mathrm{\Gamma })$. Then there exists a smooth germ $h\in {ℰ}_s$ (the ring of $C^{\mathrm{\infty }}$ germs ${\mathrm{R}}^s\mathrm{\to }\mathrm{R}$) such that $f(x)=h(u_1(x),\mathrm{\dots },u_s(x))$. [GSS]

proof:

The proof is shown on page 58 of [GSS].

theorem: (as stated by Gerald W. Schwarz)

Let $G$ be a compact Lie group acting orthogonally on ${ℝ}^n$, let ${\rho }_1,\mathrm{\dots },{\rho }_k$ be generators of $𝒫{({ℝ}^n)}^G$ (the set $G$-invariant polynomials on ${\mathrm{R}}^n$), and let $\rho =({\rho }_1,\mathrm{\dots },{\rho }_k):{ℝ}^n\to {ℝ}^k$. Then $\rho *ℰ({ℝ}^k)=ℰ{({ℝ}^n)}^G$. [SG]

proof:

The proof is shown in the following publication [SG].